376 APPENDIX: ELEMENTS OF LINEAR ALGEBRA

Conversely, ^(b^^^n is a basis with this property, then

det(M - 1 • 1E) = (^ - X)(A2 - 1) - - - (ln - X).

The proof is by induction on n. By hypothesis, there exists a vector
bn 7^0 in E which is an eigenvector for the eigenvalue !„; in other words,
u(bn) = %nbn . Let us split E into a direct sum Kbn -f V, and let p : E -* V be
the corresponding projection (A.3.4). The mapping x-+p(u(x)) is an endo-
morphism of V, and hence there exists a basis (i1? . . . , in_1) of V such that

and consequently

for suitable scalars a . We now have

If we expand the right-hand side by means of (A.6.1 .1) and use the definition
of an alternating multilinear form, we see easily that the only term which does
not vanish is

fo - A)0i2 - X) • • • Ow - A)O, ~ A)/0(619 . . . , Z>n),

and therefore det(w — A • 1E) = (//! — A)(//2 — A) • • • (^^i — X)(Xn — A). This
proves that the ^ are (except possibly for their order) the scalars Al9 . . . , An-i9
and the calculation above also establishes the second assertion of (A.6.1 0).

The matrix of u with respect to a basis satisfying the conditions of (A.6.1 0)
is said to be lower triangular.

(A.6.1 1) For each integer k > 0, we have

(A.6.1 1 .1 ) det(w* - >l • IE) = (J* - K)(X\ - X) - - - (A* - 1).

For it follows from the formulas (A.6.1 0.1) that
u\bd = fibi + a$+1 6J+1 + - - - + ag^
and the result therefore follows from (A.6.1 0).an endomorphism.